Finite Group Databasebeta

← #352473

Subgroups — #352473

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33258C211yesmaximal, characteristic
2#29988C212yesmaximal
3#20516C211yesmaximal, characteristic
4#9907C212yesmaximal
5#9622C212yesmaximal
6#9615C211yesmaximal, characteristic
7#9614C211yesmaximal, characteristic
8#8807C211yesmaximal, characteristic
9#8657C212yesmaximal
10#8645C211yesmaximal, characteristic
11#8645C211yesmaximal, characteristic
12C2×#446C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2859C2212yes—
15#2295C2211yescharacteristic
16#2101—22no—
17#2097—22no—
18#2059—22no—
19#2026—22no—
20#1948C2212yes—
21#1947C2212yes—
22#1942—21no—
23#1942—21no—
24#1942—21no—
25#1942—21no—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1834C2212yes—
29#1831C2212yes—
30#1803C2212yes—
31#1803C2212yes—
32#1771—21no—
33#1770C2211yescharacteristic
34C8×C4⋊C4—21no—
35C8×C4⋊C4C2211yescharacteristic
36#1768—21no—
37#1767C2211yescharacteristic
38#1726—21no—
39C42×C8C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#335C2211yescharacteristic
42C2×#334C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C2×#125—22no—
48C2×C4×Q8C2311yescharacteristic
49#446—22no—
50#446Q814yes—
51C8×Q8—22no—
52C8×Q8Q814yes—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—22no—
58C2×#106—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—22no—
67C2×#97—21no—
68C2×#97—21no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391D411yescharacteristic
79#391D411yescharacteristic
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389D411yescharacteristic
85#389D411yescharacteristic
86#387—21no—
87#387—21no—
88#384C2312yes—
89#384C2312yes—
90#382—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2311yescharacteristic
93#376—21no—
94C43C2311yescharacteristic
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—21no—
102#336—21no—
103#336—22no—
104#336—22no—
105#336—21no—
106#335—22no—
107#335—22no—
108#334—24no—
109C22×Q8C2×D411yescharacteristic
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118#125—82no—
119#125—82no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8C2×Q812yes—
123C4×Q8C2×Q812yes—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C42—21no—
139C2×C42—21no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42C2411yescharacteristic
147#107—42no—
148#107—42no—
149#107—42no—
150#107—42no—
151#107—82no—
152#106—82no—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8C2×Q812yes—
162C4⋊C8C2×Q812yes—
163#103—82no—
164#103—82no—
165#97—24no—
166#97—42no—
167#97—42no—
168C4×C8—24no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—41no—
172C4×C8—41no—
173C4×C8—42no—
174C4×C8C2×Q812yes—
175C4×C8C2×Q812yes—
176C4×C8#4112yes—
177C4×C8#4112yes—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—42no—
186#95—42no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8C4⋊Q812yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C2×#4111yescharacteristic
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8C22⋊Q812yes—
225C2×C8C22⋊Q812yes—
226C2×C8C22⋊Q812yes—
227C2×C8C22⋊Q812yes—
228C2×C8#13512yes—
229C2×C8#13712yes—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4C4⋊Q812yes—
252C42—22no—
253C42—22no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42C22×Q811yescharacteristic
267C42C22×Q811yescharacteristic
268C42C2×#4111yescharacteristic
269C42C2×#4111yescharacteristic
270C23C2×#12111yescharacteristic
271Q8—22no—
272Q8—22no—
273Q8—41no—
274Q8—41no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4C2×C22⋊Q811yescharacteristic
308C2×C4C2×C22⋊Q811yescharacteristic
309C2×C4C2×C4⋊Q811yescharacteristic
310C2×C4#53211yescharacteristic
311C2×C4#53311yescharacteristic
312C2×C4#53311yescharacteristic
313C2×C4C2×#13511yescharacteristic
314C2×C4C2×#13711yescharacteristic
315C2×C4#57511yescharacteristic
316C2×C4#57511yescharacteristic
317C2×C4#57611yescharacteristic
318C2×C4#57611yescharacteristic
319C2×C4#57811yescharacteristic
320C2×C4#57811yescharacteristic
321C8—22no—
322C8—22no—
323C8—22no—
324C8—42no—
325C8—42no—
326C8—82no—
327C8—82no—
328C8#47914yes—
329C22#285711yescharacteristic
330C22#303511yescharacteristic
331C22#303711yescharacteristic
332C22#305811yescharacteristic
333C22#305911yescharacteristic
334C22#306711yescharacteristic
335C22#306711yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—42no—
343C4—42no—
344C4—81no—
345C4—81no—
346C4—162no—
347C4—162no—
348C4C2×#47911yescharacteristic
349C4#307611yescharacteristic
350C4#307811yescharacteristic
351C4#308011yescharacteristic
352C4#316411yescharacteristic
353C4#316411yescharacteristic
354C4#317511yescharacteristic
355C4#317511yescharacteristic
356C2#1921911yescharacteristic
357C2#1922011yescharacteristic
358C2#1934111yescharacteristic
359C2#1934211yescharacteristic
360C2#3320011yescharacteristic
361C2#3321511yescharacteristic
362C2#3322111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#332582
#299882
#205162
#99072
#96222
#96152
#96142
#88072
#86572
#86452